{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "from scipy.optimize import root"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 49,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "2.7473062499999997e-18\n"
     ]
    }
   ],
   "source": [
    "## constant\n",
    "hbar = 6.63e-34;\n",
    "me = 1.6e-31;\n",
    "\n",
    "print(hbar**2/(me*(1e-9)**2))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 196,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x1bd1cc0d390>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "3.673727988484643e-13\n"
     ]
    }
   ],
   "source": [
    "P = 2e3*(3*np.pi/2); # larger P gives us more solution...\n",
    "KA_scan = np.linspace(-4*np.pi, 4*np.pi, 4000);\n",
    "\n",
    "plt.figure();\n",
    "plt.plot(KA_scan, np.cos(KA_scan)+(P/(KA_scan))*np.sin(KA_scan), '.b')\n",
    "plt.axhline(1);\n",
    "plt.axhline(-1);\n",
    "plt.axvline(np.pi)\n",
    "plt.xlabel('Ka');\n",
    "plt.ylabel('RHS')\n",
    "plt.title('$mV_0a/\\hbar^2 = 6\\pi$')\n",
    "plt.savefig('solving_kp_transcendental.png', dpi = 300)\n",
    "plt.show();\n",
    "\n",
    "def RHS(x):\n",
    "    return np.cos(x)+(P/(x))*np.sin(x);\n",
    "\n",
    "## roots at pi\n",
    "print(RHS(np.pi)+1)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Notes on the Transcendental Eq.\n",
    "The solutions are values for which the  value of the root func is less than 1. Cuz then we can solve the left hand side."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 90,
   "metadata": {},
   "outputs": [],
   "source": [
    "def RHS(x):\n",
    "    return np.cos(x)+(P/(x))*np.sin(x);"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 157,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x1bd2cb24978>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "## do a scan of K...\n",
    "Kguesses = np.linspace(1e-3,4*np.pi, 10000);\n",
    "band_structure = [];\n",
    "for kguess in Kguesses:\n",
    "    val = RHS(kguess);\n",
    "    if(abs(val) <1):\n",
    "        q = np.arccos(val);\n",
    "        E = kguess**2;\n",
    "        band_structure.append([q,E]);\n",
    "band_structure = np.array(band_structure);\n",
    "\n",
    "plt.figure(figsize = (5,5))\n",
    "alpha = 0.1;\n",
    "plt.plot(band_structure[:,0], alpha*band_structure[:,1], '.b', markersize = 1);\n",
    "plt.plot(-band_structure[:,0], alpha*band_structure[:,1], '.b', markersize = 1);\n",
    "\n",
    "# plt.plot(Kguesses, alpha*Kguesses**2, '.c', markersize = 0.1);\n",
    "# plt.plot(-Kguesses, alpha*Kguesses**2, '.c', markersize = 0.1);\n",
    "\n",
    "# plt.axvline(np.pi, linestyle = '--')\n",
    "# plt.axvline(-np.pi, linestyle = '--')\n",
    "plt.xlabel('qa', fontsize = 16);\n",
    "plt.ylabel('Energy', fontsize = 16)\n",
    "plt.xlim((-np.pi, np.pi))\n",
    "plt.savefig('Konig_Penny_bands.png', dpi = 300)\n",
    "plt.show();\n",
    "    \n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## wave function solutions\n",
    "When we force the determinant of the matrix to be 0, then the rows of the matrix are linearly independent with respect to each other. In that regards, once we figure out the matrix equation, we cannot resolve any further relations on the four coefficients (A,B,C,D)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 194,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "-1.1102230246251565e-16\n",
      "5.551115123125783e-17\n",
      "0.0\n"
     ]
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x1bd2cebdbe0>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "## do a scan of K...\n",
    "plt.figure(figsize = (5,10));\n",
    "Kguesses = np.linspace(1e-3,4*np.pi, 8);\n",
    "for kguess in Kguesses:\n",
    "    val = RHS(kguess);\n",
    "    if(abs(val) <1):\n",
    "        q = np.arccos(val);\n",
    "        E = kguess**2;\n",
    "        x2 = np.linspace(0,1, 100); #a has been 1 in everything we've done\n",
    "        x1 = np.linspace(-1, 0, 100);\n",
    "        xtot = np.linspace(-1, 1, 100);\n",
    "        C = 1; D = 0;\n",
    "        print(np.cos(q) - RHS(kguess))\n",
    "        A = C*np.exp(1j*q)*np.exp(1j*kguess);\n",
    "        B = D*np.exp(1j*q)*np.exp(-1j*kguess);\n",
    "        \n",
    "        ## true wavefunction reconstruction\n",
    "        psi1 = A*np.exp(1j*kguess*(x1)) + B*np.exp(-1j*kguess*(x1))\n",
    "        psi2 = C*np.exp(1j*kguess*x2) + D*np.exp(-1j*kguess*x2)\n",
    "        \n",
    "        ## check that the bloch boundary is satisfied\n",
    "        psi_check = A*np.exp(1j*kguess*(x1-1)) + B*np.exp(-1j*kguess*(x1-1))\n",
    "        psi_check_2 = ( C*np.exp(1j*kguess*x1) + D*np.exp(-1j*kguess*x1))*np.exp(1j*q)\n",
    "        \n",
    "        ## we should not be able to do this..\n",
    "        psi_test = A*np.exp(1j*kguess*(xtot)) + B*np.exp(-1j*kguess*(xtot))\n",
    "        psi_test2 =  C*np.exp(1j*kguess*xtot) + D*np.exp(-1j*kguess*xtot)\n",
    "        plt.subplot(211);\n",
    "        plt.plot(x1, psi1, '.-r');\n",
    "        plt.plot(x2, psi2, '.-b');\n",
    "        #plt.plot(xtot, psi_test, '.-b')\n",
    "        #plt.plot(xtot, psi_test2, '.-r')\n",
    "        plt.axvline(0, linestyle = '--')\n",
    "\n",
    "        plt.subplot(212)\n",
    "        plt.plot(x1, psi_check, '.y')\n",
    "        plt.plot(x1, psi_check_2, '.r', markersize = 0.9)\n",
    "\n",
    "        \n",
    "plt.show();"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## negative sign of the potential"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 201,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x1bd2ddf1898>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "def RHS_flip(x):\n",
    "    P = 10*np.pi;\n",
    "    return np.cos(x)+(P/(x))*np.sin(x);\n",
    "\n",
    "## do a scan of K...\n",
    "Kguesses = np.linspace(1e-3,4*np.pi, 10000);\n",
    "band_structure = [];\n",
    "for kguess in Kguesses:\n",
    "    val = RHS_flip(kguess);\n",
    "    if(abs(val) <1):\n",
    "        q = np.arccos(val);\n",
    "        E = kguess**2;\n",
    "        band_structure.append([q,E]);\n",
    "band_structure = np.array(band_structure);\n",
    "\n",
    "plt.figure(figsize = (5,5))\n",
    "alpha = 0.1;\n",
    "plt.plot(band_structure[:,0], alpha*band_structure[:,1], '.g', markersize = 1);\n",
    "plt.plot(-band_structure[:,0], alpha*band_structure[:,1], '.g', markersize = 1);\n",
    "\n",
    "# plt.plot(Kguesses, alpha*Kguesses**2, '.c', markersize = 0.1);\n",
    "# plt.plot(-Kguesses, alpha*Kguesses**2, '.c', markersize = 0.1);\n",
    "\n",
    "# plt.axvline(np.pi, linestyle = '--')\n",
    "# plt.axvline(-np.pi, linestyle = '--')\n",
    "plt.xlabel('qa', fontsize = 16);\n",
    "plt.ylabel('Energy', fontsize = 16)\n",
    "plt.xlim((-np.pi, np.pi))\n",
    "plt.savefig('Konig_Penny_well_bands.png', dpi = 300)\n",
    "plt.show();\n",
    "    \n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 3",
   "language": "python",
   "name": "python3"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.6.8"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 2
}
